If n=pmnsuperscriptpmn=p^{m}, where ppp is prime, then every non-zero element containing a factor of ppp is nilpotent. For example, if n=16n16n=16, then 64=0superscript6406^{4}=0.

(b)

If n=2⁢pn2pn=2p, where ppp is an odd prime, then ppp is a non-trivial idempotent element (p2=psuperscriptp2pp^{2}=p), and since 2p-1≡1(modp)superscript2p1annotated1pmodp2^{{p-1}}\equiv 1\;\;(\mathop{{\rm mod}}p) by Fermat’s little theorem, we see that a=2p-2asuperscript2p2a=2^{{p-2}} is a relative inverse of 222, as 2⋅a⋅2=2normal-⋅2a222\cdot a\cdot 2=2 and a⋅2⋅a=anormal-⋅a2aaa\cdot 2\cdot a=a

(c)

If n=2m⁢pnsuperscript2mpn=2^{m}p, where ppp is an odd prime, and m>1m1m>1, then 222 is eventually periodic. For example, n=96n96n=96, then 22=4superscript2242^{2}=4, 23=8superscript2382^{3}=8, 24=16superscript24162^{4}=16, 25=32superscript25322^{5}=32, 26=64superscript26642^{6}=64, 27=32superscript27322^{7}=32, 28=64superscript28642^{8}=64, etc…

Group with Zero. A semigroup SSS is called a group with zero if it contains a zero element 000, and S-{0}S0S-\{0\} is a subgroup of SSS. In RRR in the previous example is a division ring, then RRR with the ring multiplication is a group with zero. If GGG is a group, by adjoining GGG with an extra symbol 000, and extending the domain of group multiplication ⋅normal-⋅\cdot by defining 0⋅a=a⋅0=0⋅0:=0normal-⋅0anormal-⋅a0normal-⋅00assign00\cdot a=a\cdot 0=0\cdot 0:=0 for all a∈GaGa\in G, we get a group with zero S=G∪{0}SG0S=G\cup\{0\}.

8.

As mentioned earlier, every monoid is a semigroup. If SSS is not a monoid, then it can be embedded in one: adjoin a symbol 111 to SSS, and extend the semigroup multiplication ⋅normal-⋅\cdot on SSS by defining 1⋅a=a⋅1=anormal-⋅1anormal-⋅a1a1\cdot a=a\cdot 1=a and 1⋅1=1normal-⋅1111\cdot 1=1, we get a monoid M=S∪{1}MS1M=S\cup\{1\} with multiplicative identity111. If SSS is already a monoid with identity111, then adjoining 1′superscript1normal-′1^{{\prime}} to SSS and repeating the remaining step above gives us a new monoid with identity 1′superscript1normal-′1^{{\prime}}. However, 111 is no longer an identity, as 1′=1⋅1′superscript1normal-′normal-⋅1superscript1normal-′1^{{\prime}}=1\cdot 1^{{\prime}}.